The solution for the crossover DSW (CDSW) illustrated in Figure REF (c) will now be derived. As can be seen from
Figure REF (c), moving to the CDSW regime results in the KdV-type DSW in the PDSW and RDSW regimes becoming unstable,
with the waves in the DSW ceasing to have a near linear amplitude decrease, as seen in Figure REF , which gives
a detail view of the DSW of Figure REF (c).
The numerical leading edge velocity \(s_{+}\)  of Figure REF (b) shows that it has a maximum in \(u_{+}\)  around the boundary between the RDSW and CDSW regimes, so that the admissibility conditions discussed above give that the DSW becomes unstable, matching the numerical behaviour. Figure REF (c) shows that for a large part of the CDSW, the waves have nearly equal amplitudes, with
a rapid drop near its trailing edge. As discussed, this detailed figure also shows that the resonant wavetrain is non-uniform,
in stark contrast with the resonant DSW solution of the 5th order KdV equation, equation (REF ) with \(\mu =0\) , for
which the resonant wavetrain has a constant amplitude with a front consisting of a partial DSW which takes the wavetrain
down to the initial level \(u_{+}\)  ahead [1]}. The same DSW structure as the present CDSW was found for the
(unstable) DSW for the focusing NLS equation [2]}. The DSW for this equation consists of a train of nearly
equal amplitude waves, with a rapid decrease to the level \(u_{-}\)  behind at the trailing edge. The work of [2]} found that the major portion of unstable DSWs consists of a train of nearly equal amplitude waves,
justifying the approximate theory of [4]} which approximates DSWs as a train of equal amplitude solitary waves,
with the amplitude, width and spacing of the solitary waves found from conservation laws, motivated by the work of [5]}
on the transcritical flow of a fluid over topography. The approximate method of [4]} will then be used here
to derive a solution for the nematic CDSW.
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